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When a regression model is estimated with a dichotomous (0,1) dependent variable, the interpretation of the regression coefficients is in terms of probabilities. For a standard multiple regression equation, Yi = Σ βkXik + εi with a dichotomous dependent variable, each regression coefficient gives the expected change in the probability that Yi = 1 for a one-unit change in the predictor variable, holding all other variables constant. There are several well-known statistical problems with estimating a regression model with a dichotomous dependent variable; for example, the error term is not normally distributed and does not have a constant variance (it is heteroskedastic). The most important problem is that the linear, additive regression model may generate predicted values that are less than 0 or greater than 1, which clearly violates the requirement that those predicted values be true probabilities. The linear, additive functional form of the standard regression model is often not capable of properly modeling a dichotomous dependent variable.
There are alternative models that do satisfy the requirement that predicted values (probabilities) vary only from 0 to 1. The logit model is one of the most commonly used statistical methods for dichotomous dependent variables. The basic form of the logit model is: ln (Pi / (1 - Pi)) = Σ βkXik + εi, where Pi is the probability that Yi = 1, and ln is the logarithm to the base e (natural logarithm). Transforming the dependent variable into the natural logarithm of the ratio of Pi to 1 - Pi provides a functional form that restricts the predicted values to range from 0 to 1. The parameters of the logit model cannot be estimated via least squares as is the standard multiple regression model. However, it is relatively straightforward to use maximum likelihood methods to obtain values of the parameters that maximize the likelihood of observing a given sample of data, and many statistical software programs will do this.
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